$f(x, y, z) = x\sin(y) + y^2 + 2x + 1$ What are all the critical points of $f$ ? Choose 1 answer: Choose 1 answer: (Choice A, Incorrect) Incorrect $(0, \pi)$ (Choice B, Incorrect) Incorrect $(-1, 0)$ (Choice C, Incorrect) Incorrect $(1, 0)$ (Choice D, Checked, Correct) Correct (selected) There are no critical points.
Explanation: A critical point of a scalar field $f$ is where $\nabla f = \bold{0}$. [What's that bolded 0?] Let's find the gradient of $f$ ! $\nabla f = \begin{bmatrix} \sin(y) + 2 \\ \\ x\cos(y) + 2y \end{bmatrix}$ Notice that the $x$ -component of the gradient is always greater than $0$. In other words, no matter what input we feed the gradient of $f$, it will never equal the zero vector. Therefore, there are no critical points.